On a Generalisation of Finite T-Groups

Abstract

Let $\sigma =\{\sigma _i |i\in I\}$ be some partition of all primes ${\mathbb {P}}$ and G a finite group. A subgroup H of G is said to be $\sigma $-subnormal in G if there exists a subgroup chain $H=H_0\le H_1\le \cdots \le H_n=G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})_{H_i}$ is a finite $\sigma _j$-group for some $j \in I$ for $i = 1, \ldots , n$. We call a finite group G a $T_{\sigma }$-group if every $\sigma $-subnormal subgroup is normal in G. In this paper, we analyse the structure of the $T_{\sigma }$-groups and give some characterisations of the $T_{\sigma }$-groups.

Keywords

Finite groups / $\sigma $-groups')">$\sigma $-groups / Generalised T-groups / $\sigma $-subnormal')">$\sigma $-subnormal / ${\mathfrak {R}}_{\sigma _i}$')">The condition ${\mathfrak {R}}_{\sigma _i}$

References

[1] Al-Sharo KhA, Skiba AN. On finite groups with $\sigma $-subnormal Schmidt subgroups. Commun. Algebra. 2017, 45 4158-4165

[2] Ballester-Bolinches A, Esteban-Romero R, Asaad M. Products of Finite Groups. 2010 Berlin: Walter de Gruyter

[3] Beidleman JC, Skiba AN. On $\tau _{\sigma }$-quasinormal subgroups of finite groups. J. Group Theory. 2017, 20 5 955-964

[4] Doerk K, Hawkes T. Finite Soluble Groups. 1992 Berlin: Walter de Gruyter

[5] Gaschütz W. Gruppen, in denen das Normalteilersein transitivist. J. Reine Angew. Math.. 1957, 198 87-92

[6] Guo W. Structure Theory for Canonical Classes of Finite Groups. 2015 Heidelberg: Springer

[7] Guo W, Skiba AN. Finite groups with permutable complete Wielandt sets of subgroups. J. Group Theory. 2015, 18 191-200

[8] Guo W, Skiba AN. Groups with maximal subgroups of Sylow subgroups $\sigma $-permutably embedded. J. Group Theory. 2017, 20 1 169-183

[9] Guo W, Skiba AN. On the lattice of $\Pi _{{\mathfrak{I}}}$-subnormal subgroups of a finite group. Bull. Aust. Math. Soc.. 2017, 96 2 233-244

[10] Guo W, Skiba AN. Finite groups whose $n$-maximal subgroups are $\sigma $-subnormal. Sci. China Math.. 2019, 62 7 1355-1372

[11] Guo W, Zhang C, Skiba AN. On ${\sigma }$-supersoluble groups and one generalization of $CLT$-groups. J. Algebra. 2018, 512 92-108

[12] Hall P. Theorem like Sylow’s. Proc. Lond. Math. Soc.. 1956, 6 3 286-304

[13] Huang J, Hu B, Skiba AN. A generalisation of finite $PT$-groups. Bull. Aust. Math. Soc.. 2018, 97 3 396-405

[14] Peng TA. Finte groups with pronormal subgroups. Proc. Am. Math. Soc.. 1969, 20 232-234

[15] Peng TA. Pronormality in finite groups. J. Lond. Math. Soc.. 1971, 3 2 301-306

[16] Robinson DJS. A Course in the Theory of Groups. 1982 Heidelberg: Springer

[17] Robinson DJS. A note on finite groups in which normality is transitive. Proc. Am. Math. Soc.. 1968, 19 933-937

[18] Shemetkov, L.A.: Formation of Finite Groups. Nauka, Main Editorial Board for Physical and Mathematical Literature, Moscow (1978)

[19] Skiba AN. On $\sigma $-subnormal and $\sigma $-permutable subgroups of finite groups. J. Algebra. 2015, 436 1-16

[20] Skiba AN. Some characterizations of finite $\sigma $-soluble $P\sigma T$-groups. J. Algebra. 2018, 495 114-129

[21] Skiba AN. On sublattices of the subgroup lattice defined by formation Fitting sets. J. Algebra. 2020, 550 69-85

[22] Skiba AN. A generalization of a Hall theorem. J. Algebra Appl.. 2015, 15 4 21-36

[23] Skiba AN. On some results in the theory of finite partially soluble groups. Commun. Math. Stat.. 2016, 4 281-309

[24] Zhang C, Skiba AN. On $\sum _t^{\sigma }$-closed classes of finite groups. Ukr. Math. J.. 2018, 70 12 1707-1716

[25] Zhang C, Safonov VG, Skiba AN. On $n$-multiply $\sigma $-local formations of finite groups. Commun. Algebra. 2019, 47 3 957-968

[26] Zhang C, Wu Z, Guo W. On weakly $\sigma $-permutable subgroups of finite groups. Publ. Math. Debrecen. 2017, 91 489-502

Funding

Fundamental Research Funds for the Central Universities(2020QN20)

National Natural Science Foundation of China(12001526)